Theorem dsndxnbasendx

Description: The slot for the distance is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024) (Proof shortened by AV, 28-Oct-2024)

	Ref	Expression
Assertion	dsndxnbasendx	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\ne {\mathrm{Base}}_{\mathrm{ndx}}$$

Proof

Step	Hyp	Ref	Expression
1		basendxnn	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\in \mathbb{N}$$
2	1	nnrei	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\in ℝ$$
3		basendxltdsndx	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}<\mathrm{dist}\left(\mathrm{ndx}\right)$$
4	2 3	gtneii	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\ne {\mathrm{Base}}_{\mathrm{ndx}}$$